Properties

Label 2-528-1.1-c3-0-20
Degree $2$
Conductor $528$
Sign $-1$
Analytic cond. $31.1530$
Root an. cond. $5.58148$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3.48·5-s + 4.74·7-s + 9·9-s − 11·11-s − 15.0·13-s + 10.4·15-s + 73.1·17-s + 78.7·19-s − 14.2·21-s − 112·23-s − 112.·25-s − 27·27-s + 243.·29-s − 278.·31-s + 33·33-s − 16.5·35-s + 102.·37-s + 45.0·39-s − 241.·41-s + 280.·43-s − 31.4·45-s + 169.·47-s − 320.·49-s − 219.·51-s − 409.·53-s + 38.3·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.312·5-s + 0.256·7-s + 0.333·9-s − 0.301·11-s − 0.320·13-s + 0.180·15-s + 1.04·17-s + 0.950·19-s − 0.147·21-s − 1.01·23-s − 0.902·25-s − 0.192·27-s + 1.55·29-s − 1.61·31-s + 0.174·33-s − 0.0799·35-s + 0.454·37-s + 0.185·39-s − 0.918·41-s + 0.993·43-s − 0.104·45-s + 0.527·47-s − 0.934·49-s − 0.602·51-s − 1.06·53-s + 0.0940·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-1$
Analytic conductor: \(31.1530\)
Root analytic conductor: \(5.58148\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 528,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 3T \)
11 \( 1 + 11T \)
good5 \( 1 + 3.48T + 125T^{2} \)
7 \( 1 - 4.74T + 343T^{2} \)
13 \( 1 + 15.0T + 2.19e3T^{2} \)
17 \( 1 - 73.1T + 4.91e3T^{2} \)
19 \( 1 - 78.7T + 6.85e3T^{2} \)
23 \( 1 + 112T + 1.21e4T^{2} \)
29 \( 1 - 243.T + 2.43e4T^{2} \)
31 \( 1 + 278.T + 2.97e4T^{2} \)
37 \( 1 - 102.T + 5.06e4T^{2} \)
41 \( 1 + 241.T + 6.89e4T^{2} \)
43 \( 1 - 280.T + 7.95e4T^{2} \)
47 \( 1 - 169.T + 1.03e5T^{2} \)
53 \( 1 + 409.T + 1.48e5T^{2} \)
59 \( 1 + 196T + 2.05e5T^{2} \)
61 \( 1 + 701.T + 2.26e5T^{2} \)
67 \( 1 + 900.T + 3.00e5T^{2} \)
71 \( 1 + 756.T + 3.57e5T^{2} \)
73 \( 1 + 1.01e3T + 3.89e5T^{2} \)
79 \( 1 - 327.T + 4.93e5T^{2} \)
83 \( 1 - 756.T + 5.71e5T^{2} \)
89 \( 1 - 508.T + 7.04e5T^{2} \)
97 \( 1 - 614.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.11016588891183107888154344013, −9.254525088331411924577258089752, −7.924512373185252262401273636452, −7.46886609319762748482861277275, −6.16200721750146439473014518873, −5.34605527422431501609677021062, −4.32805515397844322074878777777, −3.10080724396107140755644242478, −1.49550935135242168299182012784, 0, 1.49550935135242168299182012784, 3.10080724396107140755644242478, 4.32805515397844322074878777777, 5.34605527422431501609677021062, 6.16200721750146439473014518873, 7.46886609319762748482861277275, 7.924512373185252262401273636452, 9.254525088331411924577258089752, 10.11016588891183107888154344013

Graph of the $Z$-function along the critical line